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UPDATE: Check out the deviation Celestial Shuriken 2 uploaded by 
and follow the links under Artist’s comments. He have found exact the same ghostlike phenomena described in this journal 
The difference is that in this case the phenomena is visible thanks to inside filter. Whiteout inside filter the central basin in Celestial Shuriken would be completely black 
Also check out the other part of his wonderful fractal gallery 
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This journal refers to some Ghostlike phenomena I have found using the Compass formula z -> z^d - da^(d-1) z (see article 27 Compasses in my Chaotic series of fractal articles).. It’s maybe the most perplexed phenomena I’ve found in the mysterious of fractals
Interesting patterns will appear when the real part of the exponent is a negative non-integer number. So let's put the exponent "d" to -3.65454654168845+0.4654564654654654i and thereafter perform a little zoom. Thus:
From GhostStart to GhostZoom3 the next zoom is shown by a rectangle. In GhostZoom4, however the next motive (the last) is appointed by an arrow. So let's go:
1) GhostStart is the parent fractal Note that small copies of the entire Mandelbrot set are visible in the periphery. Also keep in mind that the small holes that repeats in smaller and smaller scales often have the shape of the central basin.
2) GhostZoom1. At the border to the right, you see a green rectangle, denoting the zoom to the next image.
3) GhostZoom2, GhostZoom3, and GhostZoom4. We zoom towards a minibrot in the blurry region near the border of the central basin. But what! The minibrots we see under ways in that blurry region seems to be disconnected! First of all there are holes with the same shape as the central basin of the whole fractal. Second, the rest of the minibrots seems to be constituted of smaller disc-like components. That is the minibrots are more like ghostlike shadows of minibrots! This is quite obvious if you look at GhostZoom4. And, if you look at the left lower part of that image, you will see the contours of another minibrot, yet more disconnected!
But, what forms have those components building up the shadows of the minibrots? If we look at the minibrot to the right there is one big component in the body (near the neck) and one big component in the front of the head, which makes me shout. The first mentioned have the shape of a 1-periodic Julia set, and the second the shape of a 2-periodic Julia set. All this in the very way the shapes these Julia sets would have if they were drawn using parameter seeds from respective spot of the ordinary Mandelbrot set To verify this hypothesis, I made a zoom in the back of the head at the spot appointed by the green arrow, and received ..
4) ..GhostZoom5, our very last image. Her it is very clearly seen, that every black component has the shape of 2-periodic Julia sets! The speed of the 2-armed spirals increases the more near the neck (outside the image in the right upper direction) the position is located. The right part of the biggest component is cut off. That device is another phenomena in this type of fractals!
In other words: IN THIS BLURRY REGION THERE ARE SETS OF JULIA-LIKE COMPONENTS BUILDING UP GHOSTLIKE SHADOWS OF THE MANDELBROT SET! This in such a way that components of the shape of 1-periodic Julia sets produce the shape of the body, components of the shape of 2 periodic Julia sets produce the shape of the head, etc. This is a phenomena I never seen or heard of anywhere! Wonder if the great mathematicians have noticed, and maybe explained this phenomena?
The parameter files attached under Artist’s Comments refers to the formula "ExtendedCopasses" z -> z^d - da^(d-1) z + b, which means I have added a parameter “b”. In these images "b" is fixed to zero so the images could be drawn by the simpler Compass formula as well
However by this extended formula you, in the same way as in cubic parameter space obtain a four dimensional (a, b) - space, where you can glide along the non plotted axis’. Moreover you can perform rotations in the same way as describe in articles 21 and 22 in my Chaotic series
If you don't have the formula, please download the modules from here.
The above illustrations are rendered to 3200x2400 resolution (GhostBrot2 to 6400x4800) and thereafter resized in a graphical application to 1067x768 resolution. I will prepare you for a very hard work for your computer if you run the parameter files from the blurry region. Have a nice play
and follow the links under Artist’s comments. He have found exact the same ghostlike phenomena described in this journal The difference is that in this case the phenomena is visible thanks to inside filter. Whiteout inside filter the central basin in Celestial Shuriken would be completely black Also check out the other part of his wonderful fractal gallery ------------------------------------------------------------------------------
This journal refers to some Ghostlike phenomena I have found using the Compass formula z -> z^d - da^(d-1) z (see article 27 Compasses in my Chaotic series of fractal articles).. It’s maybe the most perplexed phenomena I’ve found in the mysterious of fractals
Interesting patterns will appear when the real part of the exponent is a negative non-integer number. So let's put the exponent "d" to -3.65454654168845+0.4654564654654654i and thereafter perform a little zoom. Thus:
From GhostStart to GhostZoom3 the next zoom is shown by a rectangle. In GhostZoom4, however the next motive (the last) is appointed by an arrow. So let's go:
1) GhostStart is the parent fractal Note that small copies of the entire Mandelbrot set are visible in the periphery. Also keep in mind that the small holes that repeats in smaller and smaller scales often have the shape of the central basin.
2) GhostZoom1. At the border to the right, you see a green rectangle, denoting the zoom to the next image.
3) GhostZoom2, GhostZoom3, and GhostZoom4. We zoom towards a minibrot in the blurry region near the border of the central basin. But what! The minibrots we see under ways in that blurry region seems to be disconnected! First of all there are holes with the same shape as the central basin of the whole fractal. Second, the rest of the minibrots seems to be constituted of smaller disc-like components. That is the minibrots are more like ghostlike shadows of minibrots! This is quite obvious if you look at GhostZoom4. And, if you look at the left lower part of that image, you will see the contours of another minibrot, yet more disconnected!
But, what forms have those components building up the shadows of the minibrots? If we look at the minibrot to the right there is one big component in the body (near the neck) and one big component in the front of the head, which makes me shout. The first mentioned have the shape of a 1-periodic Julia set, and the second the shape of a 2-periodic Julia set. All this in the very way the shapes these Julia sets would have if they were drawn using parameter seeds from respective spot of the ordinary Mandelbrot set
To verify this hypothesis, I made a zoom in the back of the head at the spot appointed by the green arrow, and received ..4) ..GhostZoom5, our very last image. Her it is very clearly seen, that every black component has the shape of 2-periodic Julia sets! The speed of the 2-armed spirals increases the more near the neck (outside the image in the right upper direction) the position is located. The right part of the biggest component is cut off. That device is another phenomena in this type of fractals!
In other words: IN THIS BLURRY REGION THERE ARE SETS OF JULIA-LIKE COMPONENTS BUILDING UP GHOSTLIKE SHADOWS OF THE MANDELBROT SET! This in such a way that components of the shape of 1-periodic Julia sets produce the shape of the body, components of the shape of 2 periodic Julia sets produce the shape of the head, etc. This is a phenomena I never seen or heard of anywhere! Wonder if the great mathematicians have noticed, and maybe explained this phenomena?
The parameter files attached under Artist’s Comments refers to the formula "ExtendedCopasses" z -> z^d - da^(d-1) z + b, which means I have added a parameter “b”. In these images "b" is fixed to zero so the images could be drawn by the simpler Compass formula as well
However by this extended formula you, in the same way as in cubic parameter space obtain a four dimensional (a, b) - space, where you can glide along the non plotted axis’. Moreover you can perform rotations in the same way as describe in articles 21 and 22 in my Chaotic series If you don't have the formula, please download the modules from here.
The above illustrations are rendered to 3200x2400 resolution (GhostBrot2 to 6400x4800) and thereafter resized in a graphical application to 1067x768 resolution. I will prepare you for a very hard work for your computer if you run the parameter files from the blurry region. Have a nice play
A Christmas Greeting from Sweden
As a Christmas greeting from Sweden I will promote a YouTube channel by the nature filmer and photographer, Jonna Jinton. The link goes to her channel. Go and see the 15 minutes introduction video in full screen, the image below from the beginning of the video, The scenarios are a mix from both midwinter, midsummer, and other seasons. The soundtrack is as beautiful as the scenarios, and Jonna speaks in English with a relaxing voice. I myself live in the very south of Sweden about more than thousand kilometers south from the area where Jonna and her man and dog lives, the conditions of life being quite different. Here the sun shines for about 7 hours at midwinter. Just click the introduction video, put it in full view and enjoy :snowflake: :santa: :snowflake:
How I Became Interested in Fractals
This journal ought to have been written and published for at least a decade ago The year 1987 it was an India festival in Sweden. At that time, besides my work (I was a shift worker), I was studying Sanskrit at the university of Gothenburg. My special interest was the way they wrote grammar in ancient India about 500 BC. An evening at the late autumn I was sitting with some friends in Stockholm, the capital of Sweden, and watched a TV program about Indian dance. At the end of the program, the speaker talked a little of the theory behind the dance. I don't remember what he said in details, but it was something about how the god of dance, Shiva, transformed order into chaos and chaos into order. Then a map was shown of the coast line of Great Britain and the speaker quite naturally went into that "today the mathematicians has found that there is an order behind the irregular". The subject was the new science about chaos and complexity. Both Mandelbrot and
A special flashmob
This is journal I ought to have written and published for at least one year ago. But as we say in Sweden, ”better late then never”. So first, what is a flashmob? ”A flash mob (or flashmob) is a group of people who assemble suddenly in a public place, perform for a brief time, then quickly disperse, often for the purposes of entertainment, satire, and artistic expression. Flash mobs may be organized via telecommunications, social media, or viral emails.” Quotation from Wikipedia. Back to August, 2021, when there still was some Covid restrictions in Sweden, a group around the Swedish musician and producer Christoffer Lundquist started to perform some flashmobs at some squares in Malmo (the third biggest city in Sweden) and Lund (Sweden's Oxford) at the very south of Sweden. He had just produced a new song that was a protest against the Covid restrictions we had at that time. Now remember that those restrictions were the most liberal in the western world. Yet we were less affected by
The Dreams of World Economic Forum
Hi everyone, Yuval Noah Harari is the one of the advisors to the founder and chair man of World Economic Forumm (WEF) Klaus Schwab.You can read about him (Harari) at Wikipedia.There is an interesting channel at BITCHUTE, HighImpactFlixwho has uploaded 2 videos, 9 minutes and 14 minutes where Harari dreams trans humanistic dreams, Something REALLY BAD is About to Fundamentally Change Humanity - (Pt 1) Some quotes from the video: "What should we do with all these useless people" (Already discussed, due to the fact that about 90 percent of the today's employments will disappear as a consequence of the fourth industrial revolution.) "People could look back in a hundred years and identify the corona epidemic as a moment when a new regime of surveillance took over, especially surveillance under the skin". (You probably know that you may have the vaccine passport as a chip under your skin. The proposals of vaccine passorts where already there in EU as a ”health passport”. Social
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cool!